<head>
<title>Oblique Stereographic</title>
</head>
<body>

<h1>Oblique Stereographic</h1>

<table border>

<td>Name
<td>Oblique Stereographic
<tr>

<td>EPSG Code
<td>9809
<tr>

<td>GeoTIFF Code
<td>CT_ObliqueStereographic (16)
<tr>

<td>OGC WKT Name
<td>Oblique_Stereographic
<tr>

<td>Supported By
<td>EPSG, GeoTIFF, PROJ.4
<tr>

</table>

<h3>Projection Parameters</h3>

<table border>
<th>Name
<th>EPSG #
<th>GeoTIFF ID
<th>OGC WKT
<th>Units
<th>Notes

<tr>
<td>Latitude of natural origin
<td>1
<td>NatOriginLat
<td>latitude_of_origin
<td>Angular
<td>

<tr>
<td>Longitude of natural origin
<td>2
<td>NatOriginLong
<td>central_meridian
<td>Angular
<td>

<tr>
<td>Scale factor at natural origin
<td>5
<td>ScaleAtNatOrigin
<td>scale_factor
<td>Unitless
<td>

<tr>
<td>False Easting
<td>6
<td>FalseEasting
<td>false_easting
<td>Linear
<td>

<tr>
<td>False Northing
<td>7
<td>FalseNorthing
<td>false_northing
<td>Linear
<td>

</table>

<h3>Notes</h3>

The original GeoTIFF notes didn't include a ScaleAtNatOrigin for 
Oblique Stereographic, but I have added it since it is implied by the
current EPSG tables.  See <a href="random_issues.html#stereographic">Random 
Issues</a> for some more stereographic issues.<p>

<h3>PROJ.4 Organization</h3>

The EPSG Oblique Stereographic is what is also sometimes known as
"double stereogrphic" and is distinct from the USGS (Snyder?) version
which is "proj=stere" in PROJ.4.  See 
<a href="http://bugzilla.remotesensing.org/show_bug.cgi?id=980">Bug 980</a>.<p>

<b>
<pre>
  +proj=sterea +lat_0=<i>Latitude of natural origin</i> 
               +lon_0=<i>Longitude of natural origin</i>
               +k_0=<i>Scale factor at natural origin</i> 
               +x_0=<i>False Easting</i>
               +y_0=<i>False Northing</i>
</pre>
</b>

<h3>EPSG Notes</h3>

Stereographic<p>

The Stereographic projection may be imagined to be a projection of the earth's surface onto a plane in contact with the earth at a single tangent point from the opposite end of the diameter through that tangent point. <p>

This projection is best known in its polar form and is frequently used for 
mapping polar areas where it complements the Universal Transverse Mercator 
used for lower latitudes.  Its spherical form has also been widely used by the 
US Geological Survey for planetary mapping and the mapping at small scale of 
continental hydrocarbon provinces. In its transverse or oblique ellipsoidal 
forms it is useful for mapping limited areas centred on the point where the 
plane of the projection is regarded as tangential to the ellipsoid., e.g. the 
Netherlands. The tangent point is the origin of the projected coordinate system
and the meridian through it is regarded as the central meridian. In order to 
reduce the scale error at the extremities of the projection area it is usual 
to introduce a scale factor of less than unity at the origin such that a 
unitary scale factor applies on a near circle centred at the origin and some 
distance from it. <p>

The coordinate transformation from geographical to projected coordinates is 
executed via the distance and azimuth of the point from the centre point or 
origin. For a sphere the formulas are relatively simple. For the ellipsoid the 
parameters defining the conformal sphere at the tangent point as origin are 
first derived. The conformal latitudes and longitudes are substituted for the 
geodetic  latitudes and longitudes of the spherical formulas for the origin and
the point .<p>

Oblique and Equatorial Stereographic Formula<p>

* Given the geodetic origin of the projection at the tangent point (*0, *0), 
the parameters defining the conformal sphere are:<p>

<pre>
	R= *( *0,*0)
     	n=  *[(1+e2 cos4*0)/(1-e2)]
	c=  (n+sin*0) (1-sin*0)/[(n-sin*0) (1+sin(*0)]

where:	sin*0 = (w1-1)/(w1+1)
	w1 = (S1.S2e)n
	S1 = (1+sin*0)/(1-sin*0)
	S2 = (1-e sin*0)/(1+e sin*0)
</pre>

The conformal latitude and longitude (*0,*0) of the origin are then computed from :<p>
<pre>
	*0 = sin-1[(w-1)/(w+1)]

	where S1 and S2 are as above and  w = c (S1S2e)n

 	*0  = *0
</pre>

Then for any point with geodetic coordinates (*,*) the equivalent conformal 
latitude and longitude ( * , * ) are computed from <p>

<pre>
	* = sin-1[(w-1)/(w+1)]

where 	w = c (S1S2e)n
	S1 = (1+sin*)/(1-sin*)
	S2 = (1-e.sin*)/(1+e.sin*)

and 	* = n( * - *0 ) + *0 
 
Then	B = [1+sin* sin*0 + cos*cos*0cos(* -*0 )]

and	N = FN + 2 R k0 [sin* cos*0 - cos*sin*0cos(* -*0 )] / B

	E = FE + 2 R k0 cos* sin(* -*0 ) / B
</pre>

The reverse formulae to compute the geodetic coordinates from the grid 
coordinates involves computing the conformal values, then the isometric 
latitude and finally the geodetic values.<p>

The parameters of the conformal sphere and conformal latitude and longitude 
at the origin are computed as above. Then for any point with Stereographic 
grid coordinates (E,N) :<p>

<pre>
	* = *0 + 2 tan-1[{(N-FN)-(E-FE) tan (j/2)} / (2 Rk0)]

	* = j + 2 i + *0

where	g = 2 Rk0 tan (*/4 - *0/ 2 )
	h = 4 Rk0 tan *0 + g
	i = tan-1 [(E-FE) / {h+(N-FN)}]
	j = tan-1 [(E-FE) / (g-(N-FN)] - i

Geodetic longitude 	* = (* -*0 ) / n +  *0

Isometric latitude	* = 0.5 ln [(1+ sin*) / { c (1-  sin*)}] / n

First approximation 	*1 = 2 tan-1 e*  - * / 2  where e=base of natural logarithms.

			*i = isometric latitude at *i

where			*i= ln[{tan(*i/2+* / 4}  {(1-e sin*i)/(1+e sin*i)}e/2]
 
Then iterate		*i+1 = *i - ( *i - * ) cos *i ( 1 -e2 sin2*i) / (1 - e2)

until the change in 	*   is sufficiently small.
</pre>

An alternative approach is given by Snyder, where, instead of defining a single conformal 
sphere at the origin point , the conformal latitude at each point on the ellipsoid is 
computed. The conformal longitude is then always equivalent to the geodetic longitude. 
This approach is a valid alternative to the above, but gives slightly different results away 
from the origin point.<P>

	
If the projection is the equatorial case, *0 and *0 will be zero degrees and the formulas are 
simplified as a result,but the above formulae remain valid.
<p>

For the polar version, *0 and *0 will be 90 degrees and the formulae become 
indeterminate.See below for formulae for the polar case.
<p>

For Stereographic projections centred on points in the southern hemisphere, including the 
south Polar Stereographic, the signs of E, N, *0, *,  must be reversed to be used in the 
equations and * will be negative anyway as a southerly latitude.

For Projected Coordinate System RD / Netherlands New<p>
<pre>
Parameters:
Ellipsoid   Bessel 1841    a = 6377397.155 m    1/f = 299.15281
then e = 0.08169683

Latitude Natural Origin      52o09'22.178""N  = 0.910296727 rad
Longitude Natural Origin     5o23'15.500""E  =  0.094032038 rad
Scale factor k0                 0.9999079
False Eastings FE             155000.00 m
False Northings FN           463000.00 m

Forward calculation for: 

Latitude    53oN = 0.925024504 rad
Longitude   6oE = 0.104719755 rad

first gives the conformal sphere constants:

rho0 = 6374588.71    nu0 = 6390710.613
R = 6382644.571    n = 1.000475857    c  = 1.007576465

where S1 = 8.509582274  S2 = 0.878790173  w1 = 8.428769183
sin chi0 = 0.787883237

w   = 8.492629457   chi0 = 0.909684757      D0 = d0 

for the point  chi  = 0.924394997    D = 0.104724841

hence B = 1.999870665    N = 557057.739    E = 196105.283

reverse calculation for the same Easting and Northing first gives:

g = 4379954.188    h = 37197327.96   i = 0.001102255   j = 0.008488122

then  D = 0.10472467  Longitude = 0.104719584 rad =  6 deg E

chi  = 0.924394767    psi = 1.089495123
phi1 = 0.921804948       psi1 = 1.084170164
phi2 = 0.925031162       psi2 = 1.089506925
phi3 = 0.925024504       psi3 = 1.089495505
phi4 = 0.925024504

Then Latitude      = 53o00'00.000""N
          Longitude   =   6o00'00.000
</pre>

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